Maximum principle for quasilinear SPDE's on a bounded domain without regularity assumptions

TL;DR

This paper establishes a maximum principle for local solutions of quasi-linear parabolic stochastic PDEs on bounded domains driven by space-time white noise, without requiring regularity assumptions, using an approximation method.

Contribution

It introduces a novel approach that avoids regularity assumptions for proving maximum principles in quasi-linear SPDEs on bounded domains.

Findings

Maximum principle proven for local solutions of quasi-linear SPDEs

Method does not require regularity assumptions on domain or coefficients

Results extend previous work by relaxing regularity constraints

Abstract

We prove a maximum principle for local solutions of quasi-linear parabolic stochastic PDEs, with non-homogeneous second order operator on a bounded domain and driven by a space-time white noise. Our method based on an approximation of the domain and the coefficients of the operator, does not require regularity assumptions. As in previous works, the results are consequences of It\^{o}'s formula and estimates for the positive part of local solutions which are non-positive on the lateral boundary.